Efficient Learning of Simplices

Abstract

We show an efficient algorithm for the following problem: Given uniformly random points from an arbitrary n-dimensional simplex, estimate the simplex. The size of the sample and the number of arithmetic operations of our algorithm are polynomial in n. This answers a question of Frieze, Jerrum and Kannan Frieze et al. (1996). Our result can also be interpreted as efficiently learning the intersection of n + 1 half-spaces in R^n in the model where the intersection is bounded and we are given polynomially many uniform samples from it. Our proof uses the local search technique from Independent Component Analysis (ICA), also used by Frieze et al. (1996). Unlike these previous algorithms, which were based on analyzing the fourth moment, ours is based on the third moment. We also show a direct connection between the problem of learning a simplex and ICA: a simple randomized reduction to ICA from the problem of learning a simplex. The connection is based on a known representation of the uniform measure on a simplex. Similar representations lead to a reduction from the problem of learning an affine transformation of an n-dimensional l_p ball to ICA.

Related Material

@InProceedings{pmlr-v30-Anderson13,
title = {Efficient Learning of Simplices},
author = {Joseph Anderson and Navin Goyal and Luis Rademacher},
booktitle = {Proceedings of the 26th Annual Conference on Learning Theory},
pages = {1020--1045},
year = {2013},
editor = {Shai Shalev-Shwartz and Ingo Steinwart},
volume = {30},
series = {Proceedings of Machine Learning Research},
address = {Princeton, NJ, USA},
month = {12--14 Jun},
publisher = {PMLR},
pdf = {http://proceedings.mlr.press/v30/Anderson13.pdf},
url = {http://proceedings.mlr.press/v30/Anderson13.html},
abstract = {We show an efficient algorithm for the following problem: Given uniformly random points from an arbitrary n-dimensional simplex, estimate the simplex. The size of the sample and the number of arithmetic operations of our algorithm are polynomial in n. This answers a question of Frieze, Jerrum and Kannan Frieze et al. (1996). Our result can also be interpreted as efficiently learning the intersection of n + 1 half-spaces in R^n in the model where the intersection is bounded and we are given polynomially many uniform samples from it. Our proof uses the local search technique from Independent Component Analysis (ICA), also used by Frieze et al. (1996). Unlike these previous algorithms, which were based on analyzing the fourth moment, ours is based on the third moment. We also show a direct connection between the problem of learning a simplex and ICA: a simple randomized reduction to ICA from the problem of learning a simplex. The connection is based on a known representation of the uniform measure on a simplex. Similar representations lead to a reduction from the problem of learning an affine transformation of an n-dimensional l_p ball to ICA.}
}

%0 Conference Paper
%T Efficient Learning of Simplices
%A Joseph Anderson
%A Navin Goyal
%A Luis Rademacher
%B Proceedings of the 26th Annual Conference on Learning Theory
%C Proceedings of Machine Learning Research
%D 2013
%E Shai Shalev-Shwartz
%E Ingo Steinwart
%F pmlr-v30-Anderson13
%I PMLR
%J Proceedings of Machine Learning Research
%P 1020--1045
%U http://proceedings.mlr.press
%V 30
%W PMLR
%X We show an efficient algorithm for the following problem: Given uniformly random points from an arbitrary n-dimensional simplex, estimate the simplex. The size of the sample and the number of arithmetic operations of our algorithm are polynomial in n. This answers a question of Frieze, Jerrum and Kannan Frieze et al. (1996). Our result can also be interpreted as efficiently learning the intersection of n + 1 half-spaces in R^n in the model where the intersection is bounded and we are given polynomially many uniform samples from it. Our proof uses the local search technique from Independent Component Analysis (ICA), also used by Frieze et al. (1996). Unlike these previous algorithms, which were based on analyzing the fourth moment, ours is based on the third moment. We also show a direct connection between the problem of learning a simplex and ICA: a simple randomized reduction to ICA from the problem of learning a simplex. The connection is based on a known representation of the uniform measure on a simplex. Similar representations lead to a reduction from the problem of learning an affine transformation of an n-dimensional l_p ball to ICA.

TY - CPAPER
TI - Efficient Learning of Simplices
AU - Joseph Anderson
AU - Navin Goyal
AU - Luis Rademacher
BT - Proceedings of the 26th Annual Conference on Learning Theory
PY - 2013/06/13
DA - 2013/06/13
ED - Shai Shalev-Shwartz
ED - Ingo Steinwart
ID - pmlr-v30-Anderson13
PB - PMLR
SP - 1020
DP - PMLR
EP - 1045
L1 - http://proceedings.mlr.press/v30/Anderson13.pdf
UR - http://proceedings.mlr.press/v30/Anderson13.html
AB - We show an efficient algorithm for the following problem: Given uniformly random points from an arbitrary n-dimensional simplex, estimate the simplex. The size of the sample and the number of arithmetic operations of our algorithm are polynomial in n. This answers a question of Frieze, Jerrum and Kannan Frieze et al. (1996). Our result can also be interpreted as efficiently learning the intersection of n + 1 half-spaces in R^n in the model where the intersection is bounded and we are given polynomially many uniform samples from it. Our proof uses the local search technique from Independent Component Analysis (ICA), also used by Frieze et al. (1996). Unlike these previous algorithms, which were based on analyzing the fourth moment, ours is based on the third moment. We also show a direct connection between the problem of learning a simplex and ICA: a simple randomized reduction to ICA from the problem of learning a simplex. The connection is based on a known representation of the uniform measure on a simplex. Similar representations lead to a reduction from the problem of learning an affine transformation of an n-dimensional l_p ball to ICA.
ER -